Consider a coestmer, Wetister, who may spend his income on bamultes of two gooks: first good" (f) and "aecoed good" (s). Lat ( f , s ) denote basdles of f units of the fint good and a asite of the wornd pood; both goods are avalable is cuitimusus tinerements. So, the mot of all posible bundis would be X R + 2 (the neh-argative pquadrant of two dinetisional rot epoce). Welster has uunatal preferenres; you could say he has a onetrack mind. Condider two hemdies in X 1 , x = ( f 1 , 1 ) and y = ( f 2 , 2 ) . If f 1 > f 2 then z > v rexardlese of the equantities s 1 and s 2 be wants as muxh of the first goed an breab get But if f 1 = f 2 , he peveres whichever of the baballe provides him with more of the aqood good. If f 1 f 2 and x 1 x 2 . Wetster in iadiffestat hetwera the two basulless thes appais, he in coseparing two ideutical basilis aay not be as diwious is that the Z m owe X is alis strongly manatone and stricify coinec. An informal (tat cunplete) proof is fine. Bat bark to Wetater's strangencae. To iltatrate further his ecossuricity, aupjoie that be alwass faces the folloeiag tradget eonstraint: p f f + p n whrre ( p f , p 0 ) are the price of the first good and of the mexod good, respectively, atud v is hin income. Webnter it a prie-tader in the two goods. (b) Gives his peeferences, tell ne bow much Welweter would consuse of each good given ( p y , p n , y ) . The ielation you derive abowe is an expresion for Wehster's (Marehalliad) demast for each good - we will tackle Marshallian alemand in more detail lates (or, if you ure froctactinating. by the weird about Webister 's peeference relation. And there is. His Z p oner X satisfies tationalify, strong monotonicity, abd exy strict convexity ... bat not centennityl The remaining parts of this question are meant to walk you through the argumends demosatrating that Webeter's prefatsee relation is not continuoes. To do this, you will show that there in no continoses utility fubation that can repreaeat Welister'y joeferenor relation. Let z ( f , s ) X be some abbitrary bunile. Let V ( x ) drnote a utility function eonsistent with Welses's preference relation zo over X . For out purpons, let a ntility function U be considered n + x lim U ( x 41 ) = U ( x ) (c) Let a , b > 0 . Which in greates, U ( f + a , s ) of U ( f , n + b ) ? Does the aaseer depend on the relative values of a and 6 ? Consider two sequenere. The first is { x f n = 1 , where x = f . x + n 1 } . The second is { x 2 } n = 1 where I ~ n = ( f + n 1 , x ) . Note that both sequences converge to the same bandle as n , nainely As we will want to use "proof by contradtiction," kit ws isitidly assume that U(') ts a contimous function; so assume lim n n U ( x n ) = U ( x ) and limen U ( x n ) = U ( x ) (d) Explaio who n + lim U ( 2 n ) > U ( x ) for acy positive integed . Hint, Start by arguing why V ( F + ) > V ( r ) for any two positise integrs i and f (e) Explais why U ( x 2 ) > n lim U ( e n ) for any positive integer j . Hint: Not.
I am Fabian H. I am a Calculus Homework Expert at mathsassignmenthelp.com. I hold a Master's in Mathematics, Deakin University, Australia. I have been helping students with their homework for the past 6 years. I solve homework related to Calculus.
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You can also call on +1 678 648 4277 for any assistance with Chemistry Assignments.
I am Fabian H. I am a Calculus Homework Expert at mathsassignmenthelp.com. I hold a Master's in Mathematics, Deakin University, Australia. I have been helping students with their homework for the past 6 years. I solve homework related to Calculus.
Visit mathsassignmenthelp.com or email info@mathsassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Calculus Homework.
Website Designing Company is fastest growing company in the IT market in the world for the website design and website layout. we are best website designing company in India as well as in USA we are based in Noida and Delhi NCR. Website designing company is powered by Css Founder.com
I am George P. I am a Chemistry Assignment Expert at eduassignmenthelp.com. I hold a Ph.D. in Chemistry, from Perth, Australia. I have been helping students with their homework for the past 6 years. I solve assignments related to Chemistry.
Visit eduassignmenthelp.com or email info@eduassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Chemistry Assignments.
On Application of Power Series Solution of Bessel Problems to the Problems of...BRNSS Publication Hub
One of the most powerful techniques available for studying functions defined by differential equations is to produce power series expansions of their solutions when such expansions exist. This is the technique I now investigated, in particular, its feasibility in the solution of an engineering problem known as the problem of strut of variable moment of inertia. In this work, I explored the basic theory of the Bessel’s function and its power series solution. Then, a model of the problem of strut of variable moment of inertia was developed into a differential equation of the Bessel’s form, and finally, the Bessel’s equation so formed was solved and result obtained.
In this note we generalize the regularity concept for semigroups in two ways simul- taneously: to higher regularity and to higher arity. We show that the one-relational and multi-relational formulations of higher regularity do not coincide, and each element has several inverses. The higher idempotents are introduced, and their commutation leads to unique inverses in the multi-relational formulation, and then further to the higher inverse semigroups. For polyadic semigroups we introduce several types of higher regularity which satisfy the arity invariance principle as introduced: the expressions should not depend of the numerical arity values, which allows us to provide natural and correct binary limits. In the first definition no idempotents can be defined, analogously to the binary semigroups, and therefore the uniqueness of inverses can be governed by shifts. In the second definition called sandwich higher regularity, we are able to introduce the higher polyadic idempotents, but their commutation does not provide uniqueness of inverses, because of the middle terms in the higher polyadic regularity conditions.
Math for Intelligent Systems - 01 Linear Algebra 01 Vector SpacesAndres Mendez-Vazquez
These are the initial notes for a class I am preparing for this summer in the Mathematics of Intelligent Systems. we will start with the vectors spaces, their basis and dimensions. The, we will look at one the basic applications the linear regression.
In this note we generalize the regularity concept for semigroups in two ways simul- taneously: to higher regularity and to higher arity. We show that the one-relational and multi-relational formulations of higher regularity do not coincide, and each element has several inverses. The higher idempotents are introduced, and their commutation leads to unique inverses in the multi-relational formulation, and then further to the higher inverse semigroups. For polyadic semigroups we introduce several types of higher regularity which satisfy the arity invariance principle as introduced: the expressions should not depend of the numerical arity values, which allows us to provide natural and correct binary limits. In the first definition no idempotents can be defined, analogously to the binary semigroups, and therefore the uniqueness of inverses can be governed by shifts. In the second definition called sandwich higher regularity, we are able to introduce the higher polyadic idempotents, but their commutation does not provide uniqueness of inverses, because of the middle terms in the higher polyadic regularity conditions.
MA500-2: Topological Structures 2016
Aisling McCluskey, Daron Anderson
[email protected], [email protected]
Contents
0 Preliminaries 2
1 Topological Groups 8
2 Morphisms and Isomorphisms 15
3 The Second Isomorphism Theorem 27
4 Topological Vector Spaces 42
5 The Cayley-Hamilton Theorem 43
6 The Arzelà-Ascoli theorem 44
7 Tychonoff ’s Theorem if Time Permits 45
Continuous assessment 30%; final examination 70%. There will be a weekly
workshop led by Daron during which there will be an opportunity to boost
continuous assessment marks based upon workshop participation as outlined in
class.
This module is self-contained; the notes provided shall form the module text.
Due to the broad range of topics introduced, there is no recommended text.
However General Topology by R. Engelking is a graduate-level text which has
relevant sections within it. Also Undergraduate Topology: a working textbook by
McCluskey and McMaster is a useful revision text. As usual, in-class discussion
will supplement the formal notes.
1
0 PRELIMINARIES
0 Preliminaries
Reminder 0.1. A topology τ on the set X is a family of subsets of X, called
the τ-open sets, satisfying the three axioms.
(1) Both sets X and ∅ are τ-open
(2) The union of any subfamily is again a τ-open set
(3) The intersection of any two τ-open sets is again a τ-open set
We refer to (X,τ) as a topological space. Where there is no danger of ambi-
guity, we suppress reference to the symbol denoting the topology (in this case,
τ) and simply refer to X as a topological space and to the elements of τ as its
open sets. By a closed set we mean one whose complement is open.
Reminder 0.2. A metric on the set X is a function d: X×X → R satisfying
the five axioms.
(1) d(x,y) ≥ 0 for all x,y ∈ X
(2) d(x,y) = d(y,x) for x,y ∈ X
(3) d(x,x) = 0 for every x ∈ X
(4) d(x,y) = 0 implies x = y
(5) d(x,z) ≤ d(x,y) + d(y,z) for all x,y,z ∈ X
Axiom (5) is often called the triangle inequality.
Definition 0.3. If d′ : X × X → R satisfies axioms (1), (2), (3) and (5) but
maybe not (4) then we call it a pseudo-metric.
Reminder 0.4. Every metric on X induces a topology on X, called the metric
topology. We define an open ball to be a set of the form
B(x,r) = {y ∈ X : d(x,y) < r}
for any x ∈ X and r > 0. Then a subset G of X is defined to be open (wrt the
metric topology) if for each x ∈ G, there is r > 0 such that B(x,r) ⊂ G. Thus
open sets are arbitrary unions of open balls.
Topological Structures 2016 2 Version 0.15
0 PRELIMINARIES
The definition of the metric topology makes just as much sense when we are
working with a pseudo-metric. Open balls are defined in the same manner, and
the open sets are exactly the unions of open balls. Pseudo-metric topologies are
often neglected because they do not have the nice property of being Hausdorff.
Reminder 0.5. Suppose f : X → Y is a function between the topological
spaces X and Y . We say f is continuous to mean that whenever U is open in
Y ...
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
The papers for publication in The International Journal of Engineering& Science are selected through rigorous peer reviews to ensure originality, timeliness, relevance, and readability.
Theoretical work submitted to the Journal should be original in its motivation or modeling structure. Empirical analysis should be based on a theoretical framework and should be capable of replication. It is expected that all materials required for replication (including computer programs and data sets) should be available upon request to the authors.
Consider the following sequence of operations on a Stack that begins e.docxHarrywxXPayned
Consider the following sequence of operations on a Stack that begins empty. Give the return value for each line. If it causes an error, then write ERROR.
push("A")
push("B")
pop() // FIRST
top() // SECOND
push("C")
pop() // THIRD
pop() // FOURTH
pop() // FIFTH
Group of answer choices
FIRST
[ Choose ] true false C null A B
SECOND
[ Choose ] true false C null A B
THIRD
[ Choose ] true false C null A B
FOURTH
[ Choose ] true false C null A B
FIFTH
[ Choose ] true false C null A B
.
Consider the following Joint Probability Distribution Calculate the Tr.docxHarrywxXPayned
Consider the following Joint Probability Distribution Calculate the Transinformation (Mutual Information) of the variables X and Y. Use it to check if they are independent of each other.
.
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On Application of Power Series Solution of Bessel Problems to the Problems of...BRNSS Publication Hub
One of the most powerful techniques available for studying functions defined by differential equations is to produce power series expansions of their solutions when such expansions exist. This is the technique I now investigated, in particular, its feasibility in the solution of an engineering problem known as the problem of strut of variable moment of inertia. In this work, I explored the basic theory of the Bessel’s function and its power series solution. Then, a model of the problem of strut of variable moment of inertia was developed into a differential equation of the Bessel’s form, and finally, the Bessel’s equation so formed was solved and result obtained.
In this note we generalize the regularity concept for semigroups in two ways simul- taneously: to higher regularity and to higher arity. We show that the one-relational and multi-relational formulations of higher regularity do not coincide, and each element has several inverses. The higher idempotents are introduced, and their commutation leads to unique inverses in the multi-relational formulation, and then further to the higher inverse semigroups. For polyadic semigroups we introduce several types of higher regularity which satisfy the arity invariance principle as introduced: the expressions should not depend of the numerical arity values, which allows us to provide natural and correct binary limits. In the first definition no idempotents can be defined, analogously to the binary semigroups, and therefore the uniqueness of inverses can be governed by shifts. In the second definition called sandwich higher regularity, we are able to introduce the higher polyadic idempotents, but their commutation does not provide uniqueness of inverses, because of the middle terms in the higher polyadic regularity conditions.
Math for Intelligent Systems - 01 Linear Algebra 01 Vector SpacesAndres Mendez-Vazquez
These are the initial notes for a class I am preparing for this summer in the Mathematics of Intelligent Systems. we will start with the vectors spaces, their basis and dimensions. The, we will look at one the basic applications the linear regression.
In this note we generalize the regularity concept for semigroups in two ways simul- taneously: to higher regularity and to higher arity. We show that the one-relational and multi-relational formulations of higher regularity do not coincide, and each element has several inverses. The higher idempotents are introduced, and their commutation leads to unique inverses in the multi-relational formulation, and then further to the higher inverse semigroups. For polyadic semigroups we introduce several types of higher regularity which satisfy the arity invariance principle as introduced: the expressions should not depend of the numerical arity values, which allows us to provide natural and correct binary limits. In the first definition no idempotents can be defined, analogously to the binary semigroups, and therefore the uniqueness of inverses can be governed by shifts. In the second definition called sandwich higher regularity, we are able to introduce the higher polyadic idempotents, but their commutation does not provide uniqueness of inverses, because of the middle terms in the higher polyadic regularity conditions.
MA500-2: Topological Structures 2016
Aisling McCluskey, Daron Anderson
[email protected], [email protected]
Contents
0 Preliminaries 2
1 Topological Groups 8
2 Morphisms and Isomorphisms 15
3 The Second Isomorphism Theorem 27
4 Topological Vector Spaces 42
5 The Cayley-Hamilton Theorem 43
6 The Arzelà-Ascoli theorem 44
7 Tychonoff ’s Theorem if Time Permits 45
Continuous assessment 30%; final examination 70%. There will be a weekly
workshop led by Daron during which there will be an opportunity to boost
continuous assessment marks based upon workshop participation as outlined in
class.
This module is self-contained; the notes provided shall form the module text.
Due to the broad range of topics introduced, there is no recommended text.
However General Topology by R. Engelking is a graduate-level text which has
relevant sections within it. Also Undergraduate Topology: a working textbook by
McCluskey and McMaster is a useful revision text. As usual, in-class discussion
will supplement the formal notes.
1
0 PRELIMINARIES
0 Preliminaries
Reminder 0.1. A topology τ on the set X is a family of subsets of X, called
the τ-open sets, satisfying the three axioms.
(1) Both sets X and ∅ are τ-open
(2) The union of any subfamily is again a τ-open set
(3) The intersection of any two τ-open sets is again a τ-open set
We refer to (X,τ) as a topological space. Where there is no danger of ambi-
guity, we suppress reference to the symbol denoting the topology (in this case,
τ) and simply refer to X as a topological space and to the elements of τ as its
open sets. By a closed set we mean one whose complement is open.
Reminder 0.2. A metric on the set X is a function d: X×X → R satisfying
the five axioms.
(1) d(x,y) ≥ 0 for all x,y ∈ X
(2) d(x,y) = d(y,x) for x,y ∈ X
(3) d(x,x) = 0 for every x ∈ X
(4) d(x,y) = 0 implies x = y
(5) d(x,z) ≤ d(x,y) + d(y,z) for all x,y,z ∈ X
Axiom (5) is often called the triangle inequality.
Definition 0.3. If d′ : X × X → R satisfies axioms (1), (2), (3) and (5) but
maybe not (4) then we call it a pseudo-metric.
Reminder 0.4. Every metric on X induces a topology on X, called the metric
topology. We define an open ball to be a set of the form
B(x,r) = {y ∈ X : d(x,y) < r}
for any x ∈ X and r > 0. Then a subset G of X is defined to be open (wrt the
metric topology) if for each x ∈ G, there is r > 0 such that B(x,r) ⊂ G. Thus
open sets are arbitrary unions of open balls.
Topological Structures 2016 2 Version 0.15
0 PRELIMINARIES
The definition of the metric topology makes just as much sense when we are
working with a pseudo-metric. Open balls are defined in the same manner, and
the open sets are exactly the unions of open balls. Pseudo-metric topologies are
often neglected because they do not have the nice property of being Hausdorff.
Reminder 0.5. Suppose f : X → Y is a function between the topological
spaces X and Y . We say f is continuous to mean that whenever U is open in
Y ...
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
The papers for publication in The International Journal of Engineering& Science are selected through rigorous peer reviews to ensure originality, timeliness, relevance, and readability.
Theoretical work submitted to the Journal should be original in its motivation or modeling structure. Empirical analysis should be based on a theoretical framework and should be capable of replication. It is expected that all materials required for replication (including computer programs and data sets) should be available upon request to the authors.
Consider the following sequence of operations on a Stack that begins e.docxHarrywxXPayned
Consider the following sequence of operations on a Stack that begins empty. Give the return value for each line. If it causes an error, then write ERROR.
push("A")
push("B")
pop() // FIRST
top() // SECOND
push("C")
pop() // THIRD
pop() // FOURTH
pop() // FIFTH
Group of answer choices
FIRST
[ Choose ] true false C null A B
SECOND
[ Choose ] true false C null A B
THIRD
[ Choose ] true false C null A B
FOURTH
[ Choose ] true false C null A B
FIFTH
[ Choose ] true false C null A B
.
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Consider the following Joint Probability Distribution Calculate the Transinformation (Mutual Information) of the variables X and Y. Use it to check if they are independent of each other.
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A Strategic Approach: GenAI in EducationPeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
How to Make a Field invisible in Odoo 17Celine George
It is possible to hide or invisible some fields in odoo. Commonly using “invisible” attribute in the field definition to invisible the fields. This slide will show how to make a field invisible in odoo 17.
Safalta Digital marketing institute in Noida, provide complete applications that encompass a huge range of virtual advertising and marketing additives, which includes search engine optimization, virtual communication advertising, pay-per-click on marketing, content material advertising, internet analytics, and greater. These university courses are designed for students who possess a comprehensive understanding of virtual marketing strategies and attributes.Safalta Digital Marketing Institute in Noida is a first choice for young individuals or students who are looking to start their careers in the field of digital advertising. The institute gives specialized courses designed and certification.
for beginners, providing thorough training in areas such as SEO, digital communication marketing, and PPC training in Noida. After finishing the program, students receive the certifications recognised by top different universitie, setting a strong foundation for a successful career in digital marketing.
2024.06.01 Introducing a competency framework for languag learning materials ...Sandy Millin
http://sandymillin.wordpress.com/iateflwebinar2024
Published classroom materials form the basis of syllabuses, drive teacher professional development, and have a potentially huge influence on learners, teachers and education systems. All teachers also create their own materials, whether a few sentences on a blackboard, a highly-structured fully-realised online course, or anything in between. Despite this, the knowledge and skills needed to create effective language learning materials are rarely part of teacher training, and are mostly learnt by trial and error.
Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
This webinar will introduce you to my framework, highlighting the key competencies I identified from my research. It will also show how anybody involved in language teaching (any language, not just English!), teacher training, managing schools or developing language learning materials can benefit from using the framework.
Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
Executive Directors Chat Leveraging AI for Diversity, Equity, and InclusionTechSoup
Let’s explore the intersection of technology and equity in the final session of our DEI series. Discover how AI tools, like ChatGPT, can be used to support and enhance your nonprofit's DEI initiatives. Participants will gain insights into practical AI applications and get tips for leveraging technology to advance their DEI goals.
Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
Francesca Gottschalk - How can education support child empowerment.pptxEduSkills OECD
Francesca Gottschalk from the OECD’s Centre for Educational Research and Innovation presents at the Ask an Expert Webinar: How can education support child empowerment?
Model Attribute Check Company Auto PropertyCeline George
In Odoo, the multi-company feature allows you to manage multiple companies within a single Odoo database instance. Each company can have its own configurations while still sharing common resources such as products, customers, and suppliers.
Unit 8 - Information and Communication Technology (Paper I).pdfThiyagu K
This slides describes the basic concepts of ICT, basics of Email, Emerging Technology and Digital Initiatives in Education. This presentations aligns with the UGC Paper I syllabus.
Chapter 3 - Islamic Banking Products and Services.pptx
Consider a coestmer- Wetister- who may spend his income on bamultes of.docx
1. Consider a coestmer, Wetister, who may spend his income on bamultes of two gooks: first good"
(f) and "aecoed good" (s). Lat ( f , s ) denote basdles of f units of the fint good and a asite of the
wornd pood; both goods are avalable is cuitimusus tinerements. So, the mot of all posible bundis
would be X R + 2 (the neh-argative pquadrant of two dinetisional rot epoce). Welster has uunatal
preferenres; you could say he has a onetrack mind. Condider two hemdies in X 1 , x = ( f 1 , 1 )
and y = ( f 2 , 2 ) . If f 1 > f 2 then z > v rexardlese of the equantities s 1 and s 2 be wants as
muxh of the first goed an breab get But if f 1 = f 2 , he peveres whichever of the baballe provides
him with more of the aqood good. If f 1 f 2 and x 1 x 2 . Wetster in iadiffestat hetwera the two
basulless thes appais, he in coseparing two ideutical basilis aay not be as diwious is that the Z m
owe X is alis strongly manatone and stricify coinec. An informal (tat cunplete) proof is fine. Bat
bark to Wetater's strangencae. To iltatrate further his ecossuricity, aupjoie that be alwass faces
the folloeiag tradget eonstraint: p f f + p n whrre ( p f , p 0 ) are the price of the first good and of
the mexod good, respectively, atud v is hin income. Webnter it a prie-tader in the two goods. (b)
Gives his peeferences, tell ne bow much Welweter would consuse of each good given ( p y , p n ,
y ) . The ielation you derive abowe is an expresion for Wehster's (Marehalliad) demast for each
good - we will tackle Marshallian alemand in more detail lates (or, if you ure froctactinating. by
the weird about Webister 's peeference relation. And there is. His Z p oner X satisfies tationalify,
strong monotonicity, abd exy strict convexity ... bat not centennityl The remaining parts of this
question are meant to walk you through the argumends demosatrating that Webeter's prefatsee
relation is not continuoes. To do this, you will show that there in no continoses utility fubation
that can repreaeat Welister'y joeferenor relation. Let z ( f , s ) X be some abbitrary bunile. Let V (
x ) drnote a utility function eonsistent with Welses's preference relation zo over X . For out
purpons, let a ntility function U be considered n + x lim U ( x 41 ) = U ( x ) (c) Let a , b > 0 .
Which in greates, U ( f + a , s ) of U ( f , n + b ) ? Does the aaseer depend on the relative values
of a and 6 ? Consider two sequenere. The first is { x f n = 1 , where x = f . x + n 1 } . The second
is { x 2 } n = 1 where I ~ n = ( f + n 1 , x ) . Note that both sequences converge to the same
bandle as n , nainely As we will want to use "proof by contradtiction," kit ws isitidly assume that
U(') ts a contimous function; so assume lim n n U ( x n ) = U ( x ) and limen U ( x n ) = U ( x )
(d) Explaio who n + lim U ( 2 n ) > U ( x ) for acy positive integed . Hint, Start by arguing why
V ( F + ) > V ( r ) for any two positise integrs i and f (e) Explais why U ( x 2 ) > n lim U ( e n )
for any positive integer j . Hint: Note that the smuenee { U ( x ) ) is s serictly deellinigg in that U
( x 2 ) > U ( x 2 ) for is y . (f) Now use what you have shoiWm so far to angue that m + lim U ( 2
^ n ) > n lim U ( x n ) and, therefoee, the utility function U reperenetaing Webster's preference
relation Z ove X cannot be contimasus. (Proof by Contracliction) (s) Lastly, ooasider some
atbitrary banalle x . Is there any other bundke x X macle that x f foe Wetoter? Relatt your anserer
to this part with that for (b).
